Problem: Given $ m \angle ABC = 9x + 29$, $ m \angle CBD = 8x + 23$, and $ m \angle ABD = 103$, find $m\angle ABC$. $B$ $A$ $D$ $C$
Explanation: From the diagram, we see that together ${\angle ABC}$ and ${\angle CBD}$ form ${\angle ABD}$ , so $ {m\angle ABC} + {m\angle CBD} = {m\angle ABD}$ Substitute in the expressions that were given for each measure: $ {9x + 29} + {8x + 23} = {103}$ Combine like terms: $ 17x + 52 = 103$ Subtract $52$ from both sides: $ 17x = 51$ Divide both sides by $17$ to find $x$ $ x = 3$ Substitute $3$ for $x$ in the expression that was given for $m\angle ABC$ $ m\angle ABC = 9({3}) + 29$ Simplify: $ {m\angle ABC = 27 + 29}$ So ${m\angle ABC = 56}$.